Self-similar groups: old and new results Said Najati Sidki Universidade de Brasilia In 1998 Volodya Nekrashevych and I collaborated on the paper "Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2 endomorphisms" which appeared in print in 2004. Over the past 20 years this paper stimulated the development of many ideas about self-similarity in groups, some of which are treated here.
1 Self-similarity A group G is self-similar if it is a state-closed subgroup of the automorphism group of an in…nite regular one-rooted m -tree T m ; in particular, G is residually …nite. If the action of G on the …rst level of T m is transitive we say that G is a transitive self-similar group . A group acting on the tree T m is …nite-state provided each of its elements has a …nite number of states. An automata group is a …nitely generated self-similar and …nite-state group. Self-similar and automaton representations are known for groups ranging from the torsion groups of Grigorchuk and of Gupta-Sidki to Arithmetic groups (Kapovich, 2012) and to non-abelian free groups (Glasner-Mozes, 2005; Aleshin-Vorobets, 2007). Two softwares for computation in self- similar groups are available in GAP, by Bartholdi and by Muntyan-Savchuk.
The logic of self-similar and automaton groups is com- plex. Two almost simultaneous results on unsolvability, shown in 2017: (1) P. Gillibert proved that deciding the order of an element in an automaton group unsolvable; (2) L. Bartholdi and I. Mitrofanov proved that the word problem in self-similar groups unsolvable.
2 Virtual Endomorphisms We use the notion of virtual endomorphisms to produce transitive self-similar actions. This concept often corre- sponds to contraction which had already appeared in Lie Groups and in Dynamical Systems; eg. 2 Z ! Z de…ned by 2 n 7� ! n . Given a general group G , consider a similarity pair ( H; f ) where H a subgroup of G of …nite index m and f : H ! G a homomorphism called a virtual endomorphism of G . If f is a monomorphism and the image H f is also of …nite index in G then H and H f are commensurable in G and f is a virtual automorphism . Given the pair ( H; f ) we produce by a generalized Kaloujnine- Krasner construction (abbreviated by K-K ), a transitive state-closed representation of G on the m -tree (or simply of degree m ) as follows:
let T = f t 0 (= e ) ; t 1 ; :::; t m � 1 g be a right transversal T of H in G and � : G ! Perm ( T ) be the transitive permutational representation of G on T induced from the action of the group on the right cosets of H . For each g 2 G , we obtain: (1) its image g � under �; (2) an m - tuple of elements ( h 0 ; :::; h m � 1 ) of H , called co-factors of g , de…ned by � ( t i ) g � � � 1 . h i = ( t i g ) Then, the Kaloujnine-Krasner theorem gives us a homo- morphism of G into the wreath product Hwr ( T ) G � de…ned by ' 1 : g 7! ( h i j 0 � i � m � 1) g � . This homomorphism is regarded as a …rst approximation of a representation of G on the m -ary tree. We use the virtual endomorphism f : H ! G to iterate the process in…nitely:
�� ( h i ) f � ' j 0 � i � m � 1 � g � . ' : g 7! The kernel of ' , called the f -core of H , is the largest sub- group K of H which is normal in G and is f -invariant (in the sense K f � K ). When the kernel of ' is trivial, the similarity pair ( H; f ) and f are said to be simple . A transitive state-closed group G of degree m determines a pair ( G 0 ; � 0 ) where G 0 is the stabilizer of the 0 -vertex and the projection � 0 is simple. On the other hand, a similarity pair ( H; f ) for G where [ G; H ] = m and f simple provides by the K-K construction a faithful tran- sitive state-closed representation ' of G of degree m such that [ G ' ; H ' ] = m . Problem 1 There are just two faithful transitive state- closed representations of the cyclic group G = h a i of order 2 on the binary tree a 7! � = ( e; e ) s with s the permutation (0 ; 1) and a 7! � = ( �; � ) s . On the other hand, K-K produces the unique representation a 7! � = ( e; e ) s . What is the exact relationship between self- similar representations and those produced by K-K?
3 Abelian groups Two papers by Nekrachevych-S (2004). and Brunner- S (2010) develop a fairly general study of self-similar abelian groups. Example 2 Let = Z d = h x 1 ; x 2 ; :::; x d i ; G = H h mx 1 ; x 2 ; :::; x d i ; f : mx 1 7! x 2 7! x 3 7! ::: 7! x d 7! x 1 : Then with respect to this data, G is represented as a transitive automaton group on the m -ary tree: � 1 = ( e; e; :::; e; � 2 ) �; where � = (0 ; 1 ; :::; m � 1) ; � 2 = ( � 3 ; :::; � 3 ) ; :::; � d = ( � 1 ; :::; � 1 ) . The class of abelian state-closed groups A is closed un- der topological closure and also under diagonal closure
(by adding the diagonals a z = ( a; a; :::; a ) for all a 2 A ). These facts allow exponentiation of elements of A X � i z i 2 Z 2 [ z ] which translates abelian state- by 0 � i � m closed groups language to a commutative algebra one over Z 2 . A faithful transitive self-similar representations of Z ! us- ing transcendentals in Z 2 : Theorem 3 (Bartholdi-S, 2018) Let � be a transcenden- tal unit in Z 2 . Consider the ring R = Z [1 = (2 � )] . Let G be the additive group G = R \ Z 2 and H = G \ 2 Z 2 . De…ne d : 2 Z 2 ! Z 2 by a 7! a= (2 � ) and f = d j H : Then, G is isomorphic to Z ! and the pair H ! G . ( H; f ) is simple. However there does not exist a faithful automaton representation of Z ! . Problem 4 Is there a faithful transitive self-similar rep- resentations of ( Z 2 ) ! ?
4 Nilpotent groups (with A. Berlatto, 2007) Theorem 5 Let G be a general nilpotent group, H a subgroup of …nite index m in G , f 2 Hom ( H; G ) and L = f - core ( H ) . Then, p L = f h 2 H : h n 2 L for some n g , H ker( f ) � the isolator of L in H . Denote …nitely generated torsion-free nilpotent groups of class c by T c -groups . Corollary 6 Let G be an T c -group and ( H; f ) a simple similarity pair for G . Then, f is an almost automorphism of G . In the Malcev completion of G , the virtual endo- morphism f becomes an automorphism of G .
Class 2 groups are rich in self similarity : Theorem 7 Let G be an T 2 -group and H a subgroup of …nite index in G .Then there exists a subgroup K of …nite index in H which admits a simple epimorphism f : K ! G . Given an integer m > 1 , let l ( m ) be the number of prime divisors of m (counting multiplicities) and s ( G ) the derived length of G . Theorem 8 Let G be an T c -group and H a subgroup of …nite index m in G . If f : H ! G is simple then s ( G ) � l ( m ) . There is no such bound for the nilpotency class c ( G ) :
Example 9 There exists an ascending sequence of simple triples ( G n ; H n ; f n ) where the G n ’s are metabelian 2 - generated T c -groups with [ G n ; H n ] = 4 and nilpotency class c = n . On non-existence : Problem 10 J. Dyer (1970) constructed a rational nilpo- tent Lie algebra with nilpotent automophism group. The construction yields an T c -group which does not admit a faithful transitive self-similar representation. The group is 2 -generated T 6 -group with Hirsch length 9 . Are there T 3 -groups which are not self-similar?
5 Metabelian groups Self-similar representations of metabelian groups is the next central issue of study. The following treats those of split type. Theorem 11 (Kochloukova-S) Let X be a …nitely gen- erated abelian group and B be a …nitely generated, right Z X -module of Krull dimension 1 such that C X ( B ) = f x 2 X j B ( x � 1) = 0 g = 1 . Then G = B o X admits a faithful transitive self-similar representation. The strategy of the proof : Show that there exists � in Z X such that �B is of …nite index in B and such that the map f ( �b ) = b for all b in B is core-free. De…ne the subgroup H = ( �B ) o X and extend f by f j X = id X . Then f de…nes a simple virtual endomorphism f : H ! G .
Under certain additional conditions, the group G from the above theorem is …nitely presented and of type FP m . The conditions come from the Bieri-Strebel theory of m - tame modules and its relation to the FP m -Conjecture for metabelian groups.
6 Wreath Products For wreath products of abelian groups, the guiding exam- ple is the lamplighter group G = C 2 wrC (Grigorchuk- Zuk). It has a faithful transitive self-similar representa- tion on the binary tree and is generated by s; � where s is the transposition (0 ; 1) and � = ( �; �s ) . Since we are dealing with residually …nite groups, the following result of Gruenberg is fundamental. Theorem 12 The wreath product G = BwrX is resid- ually …nite i¤ B; X are residually …nite and either B is abelian or X is …nite. Theorem 13 ( A. Dantas-S, 2017) Let G = BwrX be a transitive self-similar wreath product of abelian groups. If X is torsion free then B is a torsion group of …nite exponent. Thus, Z wr Z cannot have a faithful transitive self-similar representation.
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